The "real wholesalers require a tax ID" canard is more wishful thinking on the part of resellers than anything else. Lingerie Mart sells to both the general public and resellers; some wholesalers require a definitive reseller presence but don't actually ask for licencing; others ask for tax ID numbers as if this means anything (anyone, absolutely anyone, can get a tax ID number for free in minutes at the IRS website -- so it's hardly as if it's a serious qualification for a business.)
Simply put, there isn't really a standard for it. There are many who will insist, despite massive evidence to the contrary, that any business who doesn't require a reseller's certificate and/or tax ID are not true wholesalers, but Lingerie Mart as you point out does not require such a thing, and they definitely are a wholesale source, one of the best in their niche.
Frank